5.4: Regular and semi-regular tilings in the plane

In Problem 36 we saw that a regular n-gon has a circumcentre O. If we join each vertex to the point O, we get n triangles, each with angle sum π. Hence the total angle sum in all n triangles is πn. Since the n angles around the point O add to 2π, the angles of the regular n-gon itself have sum (n-2)π. Hence each angle of the regular n-gon has size $\left(1-\frac{2}{n}\right)\pi$ .(In the next chapter you will prove the general result that the sum of the angles in any n-gon is equal to (n-2)π radians.)

Problem 180 A regular tiling of the plane is an arrangement of identical regular polygons, which fit together edge-to-edge so as to cover the plane with no overlaps.

(a) Prove that if a regular tiling of the plane with p-gons is possible, then p = 3,4, or 6.

(b) Prove that a regular tiling of the plane exists for each of the values in (a).

We refer to the arrangement of tiles around a vertex as the vertex figure. In a regular tiling all vertex figures are automatically identical, so it is natural to refer to the tiling in terms of its vertex figure. When p = 3, exactly q = 6 tiles fit together at each vertex, and we abbreviate “six equilateral triangles” as 3⁶. In the same way we denote the tiling whose vertex figure consists of “four squares” as 4⁴, and the tiling whose vertex figure consists of “three regular hexagons” as 6³.

The natural approach in part (a) of Problem 180 is first to identify which vertex figures have no gaps or overlaps - giving a necessary condition for a regular tiling to exist. It is tempting to stop there, and to assume that this obvious necessary condition is also sufficient. The temptation arises in part because 2-dimensional regular tilings are so familiar. But it is important to recognize the distinction between a necessary and a sufficient condition; so the temptation should be resisted, and a construction given.

The procedure hidden in the solution to Problem 180 illustrates a key strategy, which dates back to the ancient Greeks, and which is called the method of analysis.

• First, we imagine that we have a typical solution to the problem - perhaps by giving it a name (even though we do not yet know anything about such a solution).
• We then use the given conditions to deduce features which any such solution must necessarily have.
• And we continue deriving more and more necessary conditions until we believe our list of derived conditions may also be sufficient.
• Finally we show that any configuration which satisfies our final derived list of necessary conditions is in fact a solution to the original problem, so that the list of necessary conditions is in fact sufficient, and we have effectively pinned down all possible solutions.

This is what we did in a very simple way in the solution to Problem 180: the condition on vertex figures gave an evident necessary condition, which turned out to be sufficient to guarantee that such a tiling exists. The same general strategy guided our classification of primitive Pythagorean triples back in Problem 23.

In the seventeenth century, this ancient Greek strategy was further developed by Fermat (1601-1665), and by Descartes (1596-1650). For example, Fermat left very few proofs; but his proof that the equation

$$x^4+y^4=z^4$$

has no solutions in positive integers x, y, z illustrated the method:

• Fermat started by supposing that a solution exists, and concluded that (x², y², z²) would then be a Pythagorean triple.
• The known formula for such Pythagorean triples then allowed him to derive even stronger necessary conditions on x, y, z.
• These conditions were so strong they could never be satisfied!

Descartes developed a “method”, whereby hard geometry problems could be solved by translating them into algebra - essentially using the method of analysis.

• Faced with a hard problem, Descartes first imagined that he had a point, or a locus, or a curve of the kind required for a solution.
• Then he introduced coordinates “x” and “y” to denote unknowns that were linked in the problem to be solved, and interpreted the given conditions as equations which the unknowns x and y would have to satisfy (i.e. as necessary constraints).
• The solutions to these equations then corresponded to possible solutions of the original problem.
• Sometimes the algebra did not quite generate a sufficient condition, giving rise to “pseudo-solutions” (values of x that satisfy the necessary conditions, but which did not correspond to actual solutions). So it was important to check each apparent solution - exactly as we did in Problem 180(b), where we checked that we can construct tilings for each of the vertex figures that arise in part (a).

The importance of the final step in this process (checking that the list of necessary constraints is also sufficient) is underlined in the next problem where we try to classify certain “almost regular” tilings.

Problem 181 A semi-regular tiling of the plane is an arrangement of regular polygons (not necessarily all identical), which fit together edge-to-edge so as to cover the plane without overlaps, and such that the arrangements of tiles around any two vertices are congruent.

(a) (i) Refine your argument in Problem 180(a) to list all possible vertex figures in a semi-regular tiling.

(ii) Try to find additional necessary conditions to eliminate vertex figures which cannot be realized, until your list of necessary conditions seems likely to be sufficient.

(b) The necessary conditions in part (a) give rise to a finite list of possible vertex figures. Construct all possible tilings corresponding to this list of possible vertex figures.

Semi-regular tilings are often called Archimedean tilings. The reason for this name remains unclear. Pappus (c. 290-c. 350 AD), writing more than 500 years after the death of Archimedes (d. 212 BC), stated that Archimedes classified the semi-regular polyhedra. Now the classification of semi-regular polyhedra (Problem 190) uses a similar approach to the classification of planar tilings, except that the sum of the angles at each vertex has sum less than (rather than exactly equal to) 360°. So it may be that the semi-regular tilings are named after Archimedes simply because he did something similar for polyhedra; or it may be that, since inequalities are harder to control than equalities, someone inferred (perhaps dodgily) that Archimedes must have known about semi-regular tilings as well as about semi-regular polyhedra. Whatever the reason, tilings and polyhedra have fascinated mathematicians, artists and craftsmen for all sorts of unexpected reasons - as indicated by:

• the fact that the classification and construction of the five regular polyhedra appear as the culmination of the thirteen books of Elements by Euclid (flourished c. 300 BC);
• the ancient Greek attempt to link the five regular polyhedra with the four elements (earth, air, fire, and water) and the cosmos;
• the ceramic tilings to be found in Islamic art - for example, on the walls of the Alhambra in Grenada;
• the book De Divina Proportions by Luca Pacioli (c. 1445-1509), and the continuing fascination with the Golden Ratio;
• the geometric sketches of Leonardo da Vinci (1452-1519);
• the work of Kepler (1571-1630) who used the regular polyhedra to explain his bold theoretical cosmology in the Astronomia Nova (1609).